

The parameter \(\beta \) , commonly referred to as the Nitsche or penalty parameter, must be chosen large enough to ensure coercivity of the bilinear form \(a_h(\cdot ,\cdot )\)  in the discrete space \(V_h\) . A sufficient condition is \(\beta > \max _{v_h \in V_{h}} \Vert  \partial _n v_h \Vert _{\partial \Omega _D}^2/\Vert  \nabla v_h \Vert _{\Omega }^2\) , which can be computed by solving a generalized eigenvalue problem [1]}. This can result in arbitrarily high values of \(\beta \)  over the entire boundary \(\partial \Omega _D\) . It is therefore common to select a local, element-wise, Nitsche parameter satisfying \(\beta |_T > \max _{v_h \in V_{h}|_T} \Vert  \partial _n v_h \Vert _{T\cap \partial \Omega _D}^2/\Vert  \nabla v_h \Vert _{T_\Omega }^2\)  by solving a local generalized eigenvalue problem for each element. Based on dimensional considerations, one can infer that \(\beta |_T\)  should be inversely proportional to a generalized thickness, \(h_{T_\Omega }\) , of \(T_\Omega \)  normal to \(T\cap \partial \Omega _D\)  [2]}, [3]}. We refer to \(h_{T_\Omega }\)  as a generalized thickness, because \(T_\Omega \)  can be irregularly shaped and, consequently, a well-defined length scale cannot generally be provided. While, with an element-wise parameter, a single small cut element does not result in a high value of \(\beta \)  over the entire boundary, an element-wise parameter is still locally unbounded, which can lead to numerical issues [4]}. With appropriate stabilization (see Section ), the properties of the immersed formulation revert to those of the boundary-fitted setting considered in the original paper by Nitsche [5]}, and a global parameter inversely proportional to the mesh width of the background mesh suffices. In the remainder of this manuscript, both element-wise and global Nitsche parameters will be indicated by \(\beta \)  and the notation \(a_h(\cdot ,\cdot )\)  and \(l_h(\cdot )\)  will be used in both stabilized and unstabilized formulations, with the choice of the Nitsche parameter following from the context. Similar to the boundary-fitted setting, the coercive and bounded bilinear form \(a_h\)  induces an equivalent operator norm \(\Vert  v_h \Vert ^2_{a_h} = a_h(v_h, v_h)\)  in the discrete space \(V_h\) . This operator norm coincides with the matrix energy norm of a corresponding coefficient vector, which renders it useful in the analysis of condition numbers of linear-algebraic systems emerging from immersed formulations.
